Solve the inequality (x^2 - 4) / (x - 1) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
B. (-∞, -2] ∪ (1, 2]
Principle or equation
To solve a rational inequality, bring all terms to one side, factor numerator and denominator, find critical points where numerator or denominator is zero, then test intervals on a sign chart. The denominator cannot be zero.
Why this answer is correct
Factor numerator: (x - 2)(x + 2) / (x - 1) ≤ 0. Critical points: x = -2, 1, 2. The denominator is zero at x = 1, so exclude x = 1. Test intervals: (-∞, -2]: positive? Choose -3: (-5)(-1)/(-4) = -5/4 negative, so include. (-2, 1): choose 0: (-2)(2)/(-1) = 4 positive, exclude. (1, 2]: choose 1.5: (-0.5)(3.5)/(0.5) = -3.5 negative, include. (2, ∞): choose 3: (1)(5)/(2) = 2.5 positive, exclude. Thus solution is (-∞, -2] ∪ (1, 2].
Example
Solve (x - 1)/(x + 2) ≤ 0: critical points -2 and 1, test intervals gives (-2, 1].
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